Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains

We consider a sequence of Dirichlet problems in varying domains(or, more generally, of relaxed Dirichlet problems involving measures in ${\mathcal M}_{0}^{+}(\Omega)$) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices $H$-converge to a matrix $A^0$, we prove that there exist a subsequence and a measure $\mu^0$ in ${\mathcal M}_{0}^{+}(\Omega)$ such that the limit problem is the relaxed Dirichlet problem corresponding to $A^0$ and $\mu^0$. We also prove a corrector result which provides an explicit approximation of the solutions in the $H^1}-norm, and which is obtained by multiplying the corrector for the $H$-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.